Teeth clenched on reading that, you can bet: many people find this band's can-do spryness, their mantle of smugness, unappealing. Still, it was an accurate precis of the route to success taken by Marcus Mumford, Ben Lovett, Winston Marshall and Ted Dwane – four London-based musicians in their 20s who came together as a folk-rock band five years ago, named themselves after an imagined high street shop, toured an awful lot and, in a hurry, became feted figures in global music.

Last autumn their second album, Babel, broke sales records in its week of release and has since gone double platinum. At Sunday night's ceremony in Los Angeles they are in contention for six Grammy awards. (They're nominated for three Brits, too, and will find out about them in a fortnight). If Mumford & Sons are one of the biggest active bands – and no act is up for more Grammy awards – then they reached this position through travel, wide and dogged. Hello Folkestone! Thank you Dixon, Illinois! Goodnight Dungog, Australia! A fanbase around the world has been carefully nurtured. Sales, airplay, awards have followed.

All bands, of course, "play live". Few seem to have quite the appetite for it as this quartet. In November, the American industry magazine Billboard deconstructed Mumford & Sons's "patient inch into the public consciousness" and concluded it was a matter of air miles: 10 separate tours of the US, crowds "courted" in big cities and at festivals but also in smaller places that others might skip. They played Bristol, Virginia, population 17,000, twice. In Portland, Maine, a grateful mayor gave them the keys to the city. "Why pick here, of all places?" an incredulous local blogger asked when they pitched up in Huddersfield last year. Mumford & Sons didn't have much of an answer, other than that they hadn't yet visited.

The band formed in December 2007, Marcus Mumford gathering musicians around him to perform songs he'd written while working as a session drummer for Laura Marling, who was then finding fame as a folk singer. (She and Mumford share a manager and were once a couple; he is now married to the actress Carey Mulligan.) "Laura's so shy, she didn't love touring that much," Mumford told me last year. "Whereas we just said yes … to all the tours."

Mumford & Sons started criss-crossing the country in a VW Polo, listening en route to a band called the Avett Brothers. This North Carolinian five-piece had gradually made it big (so Billboard would later write) by "developing their fan bases in secondary markets … touring in the corners of north America that many artists neglect." There was an early decision that Mumford & Sons would focus on America, and on its corners.

Their American push was slow to start, though, and when their first album, Sigh No More, came out in February 2010 it took six months to sell 200,000 US copies. Business was brisker in Britain at the time: when Sigh No More was nominated for the Mercury prize that year, Music Week noted that it had outsold every other album on the shortlist. Then came the 2011 Grammys: Mumford & Sons were given a coveted invitation to perform, and ended up high-fiving R Kelly (off stage) and fist-bumping Bob Dylan (on) after a brilliant three-way collaboration with Dylan and those heroes from the VW Polo days, the Avett Brothers. About 26 million people were watching.

Sigh No More quickly hit a million US copies, and by the time Babel was put out last autumn, the band had been asked to play for Obama at the White House. They'd also spawned a cottage industry. The Lumineers – a plucky folk-rock band, Denver-formed, Mumford-galvanised – are the most prominent copycats with two nominations of their own at this year's Grammys. (They'll be up against Mumford & Sons for Best Americana Album, though Mumford losing that fight would be like Daniel Day-Lewis being beaten to an Oscar by the guy who impersonates him at corporate functions.) Meanwhile, another folk-rock band, Iceland's Of Monsters and Men, told the Observer last year there was one reason they were enjoying acclaim. "Mumford & Sons. Mumford beat the path."

Strip away the 70-odd minutes a night a band spend on stage, walloped by lights and screams, and there is very little glamour to touring. Over the last two years I've ridden for hours in a tour bus with Kasabian (chief perk: unlimited crisps), killed time backstage with Foo Fighters (in a windowless cell with eight people crammed in) and waited out an afternoon in a dressing room with the xx, where a life-size cardboard cutout of Justin Bieber glared from a corner. Joining Mumford & Sons at the Hollywood Bowl last November, I found four men – sharp, funny – who were weirdly agreeable to the repetitions and indignities of touring life.

Ted Dwane, the bassist, commandeered the dressing-room bathroom to develop photographs. Lovett logged into the building's wireless network to answer business emails about a small record label he runs in London. Mumford had a Cormac McCarthy trilogy on the go, while Marshall, the banjo player, tried on a denim jacket he'd bought. Someone noticed, to his disappointment and his bandmates' delight, that it was a woman's. He wore it anyway. The band were about to play the middle show of a 60-day, 30-gig tour that started 9,000 miles west in Perth and would end 5,000 miles east in Dublin. They were completely at ease.

If a guest had climbed on to Kasabian's tour bus the afternoon I was with them, they'd have found the band sprawled in front of an episode of Police, Camera, Action!, colourfully slagging off Coldplay and eating pasties. Backstage at Mumford & Sons' Hollywood gig, a bottle of Californian red was passed around, a YouTube video of the new single by Haim was professionally critiqued, and I watched Colin Firth queue politely, for about five minutes, to get beyond a security guard and come backstage to say hello.

Those who dislike Mumford & Sons (a sizeable contingent, particularly in the UK) complain that this is exactly what's wrong with the band. They're safe. Middle class. Moreover, detractors have never been able to stomach the clothes: waistcoats, headbands, tasselled leathers – pantomime Americana. When the band were featured in GQ last year, a contradiction was bluntly announced in the headline: "Mumford & Sons are Britain's least cool band. Mumford & Sons are Britain's most successful band."

Clearly there are advantages to existing outside the boundary of cool. Mumford's music is as playable on Radio 2 as on 6Music; in the US, as Billboard explains, it's a fit for the older listeners of National Public Radio as much as for younger alt-rock stations. That the band members are Christians, religious to varying degrees with songs rich in biblical references, was never going to hurt sales. Also, the American student market – sizeable – has long thrilled to lyrics big on self-flagellation, especially those crooned in a doleful, Dave Matthews-ish growl like Mumford's. I wonder how different their career might have looked without a single couplet in the first single: "I really fucked it up this time/Didn't I my dear?" A reliable shout-along moment, particularly for earnest young men, whenever I've seen it played.

All this considered – and even factoring in the helpful TV exposure at the 2011 Grammys – America can only really be conquered by musicians who play there. Gigs the length of the country, then more gigs. Robbie Williams, having a crack at the US a decade ago, was told by Jon Bon Jovi that his band habitually played 60 to 250 shows per tour. (Williams was horrified, soon abandoning designs on the country.) That Liam Gallagher missed the first night of Oasis's US tour in the 90s to go house hunting, as Noel Gallagher recalled last year, "killed [Oasis] stone dead in America". Mumford & Sons have been constant, conscientious – safe – in their dealings with live audiences. The rewards have been huge.

When I met them before the Hollywood Bowl gig, I was put in the unusual position of telling three members of the band that the fourth – Lovett – was ailing. The keyboard player had apparently been ill during the night and had seen a doctor. "I love that I have to be told this by the fucking press," said Mumford, laughing. He didn't seem much concerned. In five years the band had cancelled only five shows, three when Mumford broke his hand. At a gig in Nashville the previous spring, he had hurried off stage, leaving the others to improvise for five minutes, then 10, while he vomited in the wings. After that, the show continued.

So Lovett went onstage at the Hollywood Bowl, green in the face. I bumped into him a few days later at the airport and he looked much better. The band were about to fly back to England for a few days' rest. Then they had gigs in Torquay, Ipswich and Llandudno.